Error analysis of the combination technique

Pflaum, Christoph; Zhou, Aihui

doi:10.1007/s002110050474

Cited by 46 publications

(38 citation statements)

References 11 publications

(22 reference statements)

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“…Let F k,l denote either P k,l or I k,l . The hierarchical surplus operator δ e is defined by Pflaum and Zhou [3] as…”

Section: Proof Of Theoremmentioning

confidence: 99%

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One dimensional combination technique and its implementation

Fang¹

2011

ANZIAMJ

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This article introduces the 1D combination technique and its implementation with parallel programming. I discuss two primary features of the 1D combination technique: (1) its reduction of computational cost, especially when combined with parallel programming and where high accuracy is required; and (2) a resultant sacrifice of accuracy. However, the loss of the accuracy can be bounded thus reducing its significance.

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“…Let F k,l denote either P k,l or I k,l . The hierarchical surplus operator δ e is defined by Pflaum and Zhou [3] as…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…The tensor product of the 1D combination technique can be used for multi-dimensional cases. For high dimensional cases, the 1D combination technique can be used together with the sparse grids technique [2,3] to increase the parallelism and reduce the parallel complexity.…”

Section: Parallel Complexity Of 1d Combination Techniquementioning

confidence: 99%

One dimensional combination technique and its implementation

Fang¹

2011

ANZIAMJ

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“…In particular, we make use of the so-called combination technique [26]. Thus, we use a standard PIDE solver on a series of full, regular, but anisotropic grids.…”

Section: Numerical Pide Solutionmentioning

confidence: 99%

Option Pricing in Hilbert Space-Valued Jump-Diffusion Models Using Partial Integro-Differential Equations

Hepperger¹

2010

SIAM J. Finan. Math.

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Abstract. Hilbert space-valued jump-diffusion models are employed for various markets and derivatives. Examples include swaptions, which depend on continuous forward curves, and basket options on stocks. Usually, no analytical pricing formulas are available for such products. Numerical methods, on the other hand, suffer from exponentially increasing computational effort with increasing dimension of the problem, the "curse of dimension." In this paper, we present an efficient approach using partial integro-differential equations. The key to this method is a dimension reduction technique based on a Karhunen-Loève expansion, which is also known as proper orthogonal decomposition. Using the eigenvectors of a covariance operator, the differential equation is projected to a low-dimensional problem. Convergence results for the projection are given, and the numerical aspects of the implementation are discussed. An approximate solution is computed using a sparse grid combination technique and discontinuous Galerkin discretization. The main goal of this article is to combine the different analytical and numerical techniques needed, presenting a computationally feasible method for pricing European options. Numerical experiments show the effectiveness of the algorithm.

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“…The resulting representation of the covariance is known as the combination technique [14]. Nevertheless, in difference to [14,28,32,42], this representation is a consequence of the Galerkin method in the sparse tensor product space and is not an additional approximation step.…”

Section: Introductionmentioning

confidence: 99%

Combination Technique Based Second Moment Analysis for Elliptic PDEs on Random Domains

Harbrecht

Peters

2016

Lecture Notes in Computational Science and Engineering

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In this article, we propose the sparse grid combination technique for the second moment analysis of elliptic partial differential equations on random domains. By employing shape sensitivity analysis, we linearize the influence of the random domain perturbation on the solution. We derive deterministic partial differential equations to approximate the random solution's mean and its covariance with leading order in the amplitude of the random domain perturbation. The partial differential equation for the covariance is a tensor product Dirichlet problem which can efficiently be determined by Galerkin's method in the sparse tensor product space. We show that this Galerkin approximation coincides with the solution derived from the combination technique provided that the detail spaces in the related multiscale hierarchy are constructed with respect to Galerkin projections. This means that the combination technique does not impose an additional error in our construction. Numerical experiments quantify and qualify the proposed method.

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Error analysis of the combination technique

Cited by 46 publications

References 11 publications

One dimensional combination technique and its implementation

One dimensional combination technique and its implementation

Option Pricing in Hilbert Space-Valued Jump-Diffusion Models Using Partial Integro-Differential Equations

Combination Technique Based Second Moment Analysis for Elliptic PDEs on Random Domains

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